imMinkowski

The imMinkowski module contains several functions for measuring or estimating geometric quantities from 2D or 3D images.

For 2D images, parameters are the area, the perimeter and the (2D) Euler Number. For 3D images, parameters are the volume, the surface area (called surface), the mean breadth (also known as integral of mean curvature), and the (3D) Euler Number. For the sake of completeness, parameters for 1D images are also included: length and number (1D Euler Number).

Several functions are provided for each parameter, depending on how the parameter is considered:

• im<Param>: global measure of the parameter in the whole image. If the structure touches the border of the image, it is considered as a structure border. Such functions should fit most needs.
• im<Param>Estimate: estimation of the parameter by considering image is a representative window of a larger structure. The intersection of the structure with image border is not taken into account for measurements.
• im<Param>Density: same as im<Param>Estimate, but the result is normalised by the area or volume of the observed window.
• im<Param>Lut: returns a look-up-table of values that can be used to estimate the parameter from the histogram of binary configurations in original image, as computed by function imBinaryConfigHisto.

Most functions work both for binary and label images. It is possible to specify options (connectivity for Euler Number, number of directions for perimeter or surface area), as well as image resolution in each direction.

Examples

% compute perimeter of several coins 
lbl = bwlabel(imread('coins.png') > 100);
imPerimeter(lbl)
ans = 
    184.8668
    154.9495
    185.1921
    267.1690
    187.3183
    179.5038
    182.7406
    180.8445
    155.5049
    155.5049

% Surface area measured in 3D binary image (result in pixel^2)
img = analyze75read(analyze75info('brainMRI.hdr'));
bin = imclose(img>0, ones([5 5 3]));
S = imSurface(bin, [1 1 2.5])      % specify resolution
ans = 
      2.7291e+004

Perimeter in 2D images

imPerimeter           - Perimeter of a 2D image using Crofton formula
imPerimeterDensity    - Perimeter density of a 2D binary structure, using Crofton formula
imPerimeterEstimate   - Perimeter estimate of a 2D binary structure
imPerimeterLut        - Look-Up Table for measuring perimeter in a binary image

Area in 2D images

imArea                - Compute area of binary 2D image 
imAreaDensity         - Compute area density in a 2D image
imAreaEstimate        - Estimate area of binary 2D structure with edge correction

Euler-Poincare characteristic in 2D images

imEuler2d             - Euler number of a binary 2D image
imEuler2dDensity      - Euler density in a 2D image
imEuler2dEstimate     - Estimate Euler number in a 2D image

Volume in 3D images

imVolume              - Volume measure of a 3D binary structure
imVolumeDensity       - Compute volume density of a 3D image
imVolumeEstimate      - Estimate volume of a 3D binary structure with edge correction

Surface area in 3D images

imSurface             - Surface area of a 3D binary structure
imSurfaceDensity      - Surface area density of a 3D binary structure
imSurfaceEstimate     - Estimate surface area of a binary 3D structure
imSurfaceLut          - Look-Up Table for computing surface area of a 3D binary image
imJointSurface        - Surface area of the interface between two labels

Mean breadth (integral of mean curvature) in 3D images

imMeanBreadth         - Mean breadth of a 3D binary or label image
imMeanBreadthDensity  - Mean breadth density of a 3D binary structure
imMeanBreadthEstimate - Estimate mean breadth of a binary structure
imMeanBreadthLut      - Look-Up Table for computing mean breadth of a 3D image

Euler-Poincare characteristic in 3D images

imEuler3d             - Euler number of a binary 3D image
imEuler3dDensity      - Compute Euler density in a 3D image
imEuler3dEstimate     - Estimate Euler number in a 3D image

Euler-Poincare characteristic and length in 1D images

imEuler1d             - Compute Euler number of a binary 1D image
imEuler1dEstimate     - Compute Euler number of a binary 1D image
imLength              - Compute total length of a binary 1D structure
imLengthDensity       - Estimate length density of a binary 1D structure using edge correction
imLengthEstimate      - Estimate total length  of a binary 1D structure using edge correction

Utility functions

imBinaryConfigHisto   - Histogram of binary configuration from 2D/3D image
imProjectedDiameter   - Projected diameter in a given direction
imProjectedArea       - Total projected area in a given direction

References

If you use this package, please be kind to cite following reference:

• David Legland, Kiên Kiêu and Marie-Françoise Devaux (2007). Computation of Minkowski measures on 2D and 3D binary images. Image Analysis and Stereology, Vol 26(2), June 2007 web: http://www.ias-iss.org/ojs/IAS/article/view/811

Following reference can also be of interest:

• Joachim Ohser and Frank Muecklich (2000) Statistical Analysis of Microstructures in Material Sciences, John Wiley and Sons.

See also the imMeasures and the imGeodesics modules, for complementary descriptive functions.